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### Uniform Temperature Differential Frame Deflections Equations and Calculator

Beam Deflection and Stress Equation and Calculators

Frame Deflections with Uniform Temperature Differential Between Outside and Inside Equations and Calculator.

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General reaction and deformation expressions with right and left ends pinned

Horizontal Deflection at A:

Reaction locations are pinned therefore, the displacements = 0 = δHA

Angular Rotation at A:

Where:

Loading Terms LPH and LPM are given below.
Reaction loads and moments VA and VB, and HB can be evaluated from equilibrium equations after calculating HA and MA.

Note: t`; t2; and t3 are beam thicknesses from inside to outside

Reaction locations are pinned therefore, the Moments = 0 = MA = MB

Where:

Δo = Displacement (in, mm),
θo = Angular Displacement (radians),
W = Load or Force (lbsf, N),
w = Unit Load or force per unit length (lbs/in2, N/mm2),
Mo = Applied couple (moment) ( lbs-in, N-mm),
θo = Externally created angular displacement (radians),
Δo, = Externally created concentrated lateral displacement (in, mm),
T1 - T2 = Uniform temperature rise (°F),
To = Average Temperature (deg °F),
γ = Temperature coefficient of expansion [ µinch/(in. °F), µmm/(mm. °F) ],
T1, T2 = Temperature on outside and inside respectively (degrees),
HA, HB = Horizontal end reaction moments at the left and right, respectively, and are positive clockwise (lbs, N),
I1, I2, and I3 = Respective area moments of inertia for bending in the plane of the frame for the three members (in4, mm4),
E1, E2, and E3 = Respective moduli of elasticity (lb/in2, Pa) Related: Modulus of Elasticity, Yield Strength;
γ1, γ2, and γ3 = Respective temperature coefficients of expansions unit strain per. degree ( in/in/°F, mm/mm/°C),
l1, l2, l3 = Member lengths respectively (in, mm),

References:

Roark's Formulas for Stress and Strain

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